Find the exact area of the surface obtained by rotating the given curve about the $x$-axis.
$$x = 2 \cos ^ { 3 } \theta , \quad y = 2 \sin ^ { 3 } \theta , \quad 0 \leq \theta \leq \pi / 2$$
A) $\frac { 24 \pi } { 5 }$
B) $\frac { 18 \pi } { 5 }$
C) $\frac { 12 \pi } { 5 }$
D) $\frac { 2 \pi } { 4 }$
E) None of these

Question

Quizplus · Accepted Answer

The curve is a parametric form of an astroid. The surface area of revolution about the x-axis is calculated using the formula $2\pi \int y \sqrt{\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2} \, d\theta$. Evaluating this integral from $\theta = 0$ to $\theta = \pi/2$ gives $\frac{24\pi}{5}$.

Find the Exact Area of the Surface Obtained by Rotating xxx

Find the Exact Area of the Surface Obtained by Rotating $x$